We leverage the Kenmochi condition 6.58 in [1]} to bound \( d \hat{\hspace{-1.49994pt}l }_\rho (\operatorname{epi}f, \operatorname{epi}\tilde{f}^\nu )\)  and proceed in two steps. First, suppose that \((u,x)\in \mathbb {R}^s\times \mathbb {R}^n\)  satisfies \(\Vert u\Vert _2 \le \rho \) , \(\Vert x\Vert _2 \le \rho \) , and \(f(u,x) \le \rho \) . Then, \(u=0\) .
Set
\(\eta _1^\nu = \max \big \lbrace {\raisebox {1pt}{\frac{1}{2}}}\theta ^\nu \Vert p^\nu -p\Vert _2^2 + \Vert y^\nu \Vert _2 \Vert p^\nu -p\Vert _2, ~\Vert p^\nu -p\Vert _2\big \rbrace .\) 
